the economics of cultural transmission and the dynamics of ... · thierry verdier delta, paris,...

Journal of Economic Theory 97, 298�319 (2001)

The Economics of Cultural Transmission and theDynamics of Preferences1

Alberto Bisin

Department of Economics, New York University, 269 Mercer Street,New York, New York 10003

alberto.bisin�nyu.edu

and

Thierry Verdier

DELTA, Paris, ENS, 48 Boulevard Jourdan, 75014 Paris, Franceverdier�delta.ens.fr

Received April 7, 1999; final version received April 14, 2000;published online January 25, 2001

This paper studies the population dynamics of preference traits in a model of inter-generational cultural transmission. Parents socialize and transmit their preferences totheir offspring, motivated by a form of paternalistic altruism (``imperfect empathy''). Insuch a setting we study the long run stationary state pattern of preferences in the popu-lation, according to various socialization mechanisms and institutions, and identifysufficient conditions for the global stability of an heterogenous stationary distributionof the preference traits.

We show that cultural transmission mechanisms have very different implicationsthan evolutionary selection mechanisms with respect to the dynamics of the distri-bution of the traits in the population, and we study mechanisms which interactevolutionary selection and cultural transmission. Journal of Economic LiteratureClassification numbers: D10, I20, J13. � 2001 Academic Press

1. INTRODUCTION

This paper studies the population dynamics of the distribution of preferencesin a model in which preference traits are endogenously determined by a processof intergenerational transmission of traits.

The view that preferences, norms, and, more generally, cultural attitudesshould be considered as endogenous with respect to socioeconomic systems

doi:10.1006�jeth.2000.2678, available online at http:��www.idealibrary.com on

2980022-0531�01 �35.00Copyright � 2001 by Academic PressAll rights of reproduction in any form reserved.

1 Thanks to Gary Becker for his comments and support. Thanks also to Jess Benhabib,Samuel Bentolila, Michele Boldrin, Tillman Borgers. Daniel Cohen, Peter Diamond, YawNiarko, Efe Ok, Arthur Robson, Sherwin Rosen, Gilles Saint Paul, Carlo Scarpa and to theparticipants to the various seminars given on the paper. We are also grateful to one anonymousreferee and an associate editor of this journal for very useful remarks and suggestions. MarcosChamon provided excellent research assistance.

has been now extensively motivated. In particular, various pieces of empiricalevidence have been interpreted to suggest the relevance of the endogeneityof various elements of preferences, as, for example, the discount factor, theperceived importance of education, the interdependence of agents' con-sumption or production patterns, and the relevance of ethnic and religiousvalues (see Borjas [12]; Duesenberry [18]; Kapteyn et al. [26]; Iannaccone[25]; Leibenstein [28]; Pollak [29]). Also, Cavalli-Sforza and Feldman[15], among others, document the dependence of children's preferences onthose of their parents.

Most analyses of the dynamics of preferences concentrate on evolutionaryselection mechanisms. In such environments, preferences are either inheritedby genetic transmission, or else are determined by an imitation process. Ineither case the resulting mechanism for the transmission of preferences ismonotonically increasing in the material economic payoff associated toeach preference trait (as the reproductive success of each trait is naturallyassumed increasing in material payoffs��see, for instance, Bester and Guth[8]; Eshel et al. [19]; Fershtman and Weiss [20]; Kockesen et al. [27];and Robson [30].2

We instead study models in which preferences of children are acquiredthrough an adaptation and imitation process which depends on theirparents' socialization actions, and on the cultural and social environmentin which children live. We assume that the parents' socialization decision ismotivated by their evaluation of their children's actions. Such evaluation isconstructed on the basis of a form of paternalistic altruism in which parentsevaluate their children's actions with their own (the parents') preferences.As a consequence, each parent always attempts to socialize his�her childrento his�her own preference trait.

Since, in our analysis, the intergenerational transmission of cultural traitsinvolves economic decisions of rational agents (the cultural parents), thetransmission mechanism is not necessarily monotonic in material payoffs,as it depends on the parents' altruistic evaluation of their children's actions.3

299CULTURAL TRANSMISSION

2 A different approach to the endogeneity of preferences concentrates on the introspectivechoice of one's own preferences, as in most of Becker's work (e.g., Becker [5], and Becker andMulligan [6]).

A large and established literature, of course, centers on the evolutionary selection of behavior,as opposed to preferences, in strategic environments (see Samuelson [33]; Vega Redondo [37];and Weibull [38] for book-length presentations).

3 An explicit analysis of the transmission and adoption of cultural traits in which thetransmission mechanism is exogenously specified (i.e., independently of any of the parents' deci-sion) has been developed by Cavalli-Sforza and Feldman [15], and Boyd and Richerson [13](see also Bandura and Walters [2] and Baumrind [3]. This literature has roots in evolutionarysociobiology (see Hamilton [22]; Dawkins [17]; Wilson [39]; and, for economic applications,Hirshleifer [23], [24]; Frank [21]; Becker [4]; Stark [35]; Bergstrom [7]; and Rubin and Paul[32].

We study the population dynamics of the distribution of preferencesinduced by such cultural transmission mechanisms. The main question weaddress is: What are the conditions on the transmission mechanisms whichinduce heterogeneity in the long run stationary distribution of preferencesin the population? Our main result, in this respect, is the following. If thedirect socialization of children inside the family and their cultural adapta-tion and imitation from society at large operate as substitutes in the wholecultural transmission mechanism, then there exists an heterogeneous distri-bution of preferences in the population, which is globally stable. Whenfamily and society are substitutes in the transmission mechanism, in fact,families will socialize children more intensely whenever the set of culturaltraits they wish to transmit is common only to a minority of the popula-tion; and, on the contrary, families which belong to a cultural majority willnot spend much resources directly socializing their children, since theirchildren will adopt or imitate with high probability the cultural trait mostpredominant in society at large, which is the one their parents desire for them.

Since the transmission mechanisms we analyze involve agents' socializa-tion decisions explicitly, it is of interest to study their welfare properties.We show that the paths of cultural evolution induced by such endogenoussocialization and transmission mechanisms are inefficient, in the sense thattoo many resources are individually invested by parents to affect thepreferences of their children.

The dynamics of the pattern of preferences obtained through endogenouscultural transmission mechanisms contrasts, in general, with those whichwould be obtained in evolutionary selection models. In simple evolutionaryenvironments in which agents do not interact (strategically or otherwise),material payoffs, and hence fitness, can only be exogenously associated toeach preference trait. Trivially, evolutionary selection models, in such environ-ments, induce population dynamics which converge to an homogeneousdistribution of preferences. Such models, in fact, lack the endogenous dif-ferential socialization success of cultural minorities, which drives thecultural heterogeneity of the stationary state distribution induced bycultural transmission mechanisms. But even if we allow the agents somechoice of reproductive pattern (endogenous fitness), families in a culturalmajority will choose relatively high fertility rates, since in this case theirchildren will inherit their trait with high probability (children are of highexpected ``quality''). The choice of reproduction patterns, as a consequence,differently from the socialization choice, induces a culturally homogeneousconfiguration of the distribution of traits in the population (which con-figuration depends on the parameters of the transmission process, includingthe initial distribution of traits).

Given these differences, it is then interesting to study what happens whenwe extend our framework to the case in which cultural transmission interacts

300 BISIN AND VERDIER

with evolutionary selection mechanisms. We show that our main implicationthat cultural substitutability generates long run cultural heterogeneity isrobust to the simultaneous introduction of differential (exogenous andendogenous) fertility dynamics.

Our paper is related to a small literature investigating the issue of culturaltransmission in dynamic models of preference evolution. Cavalli-Sforza andFeldman [16] and Boyd and Richerson [13], in their seminal work inevolutionary anthropology, were the first to propose models of cultural trans-mission with exogenous socialization efforts. Stark [35] has applied theirframework to the evolution of altruism. Bisin and Verdier [10] investigatethe interaction between cultural transmission and marital segregation, wheremarital segregation operates as a special form of a socialization mechanism,and show that such a mechanism favors, in general, heterogeneity of the dis-tribution of the population with respect to the cultural traits. The analysis isfocused on providing and discussing empirical implications for the dynamicsof marriage homogamy rates and cultural heterogeneity, with specificreference to religious and ethnic traits. The present paper is broader in itsfocus, aiming at identifying general properties of socialization mechanismswhich sustain heterogeneous limit distributions of preference traits in thepopulation, even in environments in which socialization mechanisms interactwith evolutionary selection in the process of preference formation.

The structure of the paper is as follows. Section 2 develops and analyzesa simple economic model of cultural transmission in a two-preference traitpopulation. It studies the induced population dynamics of the distributionof traits. The crucial role of substitutability and complementarity betweendirect socialization and adoption from society at large as cultural transmis-sion mechanisms is identified and analyzed. We also illustrate the analysiswith various examples of interacting socialization technologies (Section 2.2.2-3).Section 3 presents a normative discussion of cultural transmission mechanisms.Section 4 compares our approach to cultural transmission to variousevolutionary selection mechanisms and interacts cultural and evolutionarymechanisms. Finally, Section 5 concludes.

2. A SIMPLE MODEL OF CULTURAL TANSMISSION ANDPREFERENCE EVOLUTION

In this section we study a simple, economic model of cultural evolutionin a two-cultural trait population of individuals.4 We start first by describing


4 The gist of the analysis is preserved in environments with N-traits. An Appendix availablefrom the authors develops this point.

the class of socialization mechanisms which underlie our process ofpreference formation.

2.1. A Class of Socialization Mechanisms

We model the transmission of cultural traits as a mechanism which inter-acts socialization inside the family and socialization outside the family, insociety at large, via imitation and learning from particular role models liketeachers, peers etc. (Socialization inside the family is also called ``directvertical'' socialization, while socialization by society, is also called ``oblique''socialization5). Suppose there are two possible types of cultural traits in thepopulation, [a, b]. The fraction of individuals with trait i # [a, b] is denotedqi. Different traits might correspond, for instance, to different perceptionsof risk, different time preferences, or more generally different forms ofutility functions. Families are composed of one parent and a child, andhence reproduction is asexual.6 All children are born without definedpreferences or cultural traits and are first exposed to their parent's trait.``Direct vertical'' socialization to the parent's trait, say i, occurs with prob-ability d i (qi).7 If a child from a family with trait i is not directly socialized,which occurs with probability 1&d i (qi), he or she picks the trait of a rolemodel chosen randomly in the population (i.e., he or she picks trait i withprobability qi and trait j =3 i with probability q j=1&q i).

Let Pij denote the probability that a child from a family with trait i issocialized to trait j. By the Law of Large Numbers Pij will also denote thefraction of children with a type i parent who have preferences of type j.8

The socialization mechanism just introduced is then characterized by thefollowing transition probabilities, for all i, j # [a, b]:

Pii=d i (q i)+(1&d i (qi)) qi (1)

Pij=(1&d i (qi))(1&qi). (2)


5 This terminology is taken by Cavalli-Sforza and Feldman [16].6 See Section 2 for the extension to endogenous fertility.7 The probability of direct socialization is allowed to depend on the fraction of individuals

with trait i in the population, qi, to capture a possible interaction between the direct verticaltransmission of traits by the family and the distribution of traits in the population. Such inter-actions will be crucial in the models of the next section in which direct socialization constitutesan endogenous choice of the parent.

8 See Al-Najjar [1] and Sun [36] for formal constructions of the Law of Large Numberswith a continuum of independent agents.

The dynamics of the fraction of the population with trait i, in thecontinuous time limit,9 is then characterized by:

q* i=qi (1&q i)[d i (q i)&d j (1&qi)]. (3)

This is the basic equation for the dynamics of cultural traits analyzed in thepaper. It is easy to derive conditions which guarantee that the distributionof cultural traits in the population will converge to a non-degeneratedistribution, i.e., such that some heterogeneity of traits is maintained in thelimit population. We turn to this point next.

An important determinant of the process of cultural transmission in theanalysis of the population dynamics of the different cultural traits consistsin how the external environment, reflected by qi, affects direct verticalsocialization. More precisely, situations where the social environment actsas a substitute or as a complement to family cultural transmission technologieshave very different implications for the dynamics of cultural traits. A precisedefinition of ``substitution'' is useful.

Definition (Cultural substitution). Vertical cultural transmissionand oblique cultural transmission are cultural substitutes for agent i (or,equivalently, d i (qi) satisfies the cultural substitution property) if

d i (qi) is a continuous, strictly decreasing function in qi, and, moreover, d i (1)=0.

Intuitively, we say that direct vertical transmission acts as a culturalsubstitute to oblique transmission whenever parents have less incentives tosocialize their children the more widely dominant are their values in thepopulation (in the limit of a perfectly homogenous populations of type i,whenever parents of type i do not directly socialize their children). We nowcharacterize the dynamics of (3) under cultural substitution (see alsoFig. 1). Let qi (t, q i

0) denote the path of qi solving (3) under initial conditionqi (0)=q i

0 .


9 The continuous time approximation can be derived, for instance, from an economy withoverlapping generations of agents living 2 units of time and having children 1&h units oftime after birth, by taking the limit for 2, h � 0, with h�2 � 0. For a related continuous timeapproximation in evolutionary models, see, for instance, Cabrales�Sobel [14].

It is easy to see that our results hold in the discrete time dynamics in the following sense:The global stability results of the form qi (t, q i

0) � q i* for any initial population shareqi

0 # (0, 1), in the continuous time dynamics (in Proposition 1, 3, 4, and 5), correspond, in thediscrete time dynamics, to the existence of a neighborhood of qi *, strictly contained in (0, 1),which is globally attracting, for any initial population share q i

0 # (0, 1).

FIG. 1. The dynamics under cultural substitution.

Proposition 1. Suppose vertical and oblique cultural transmission arecultural substitutes for both groups a and b. Then, (0, 1, qi *) are the station-ary states of (3), and 0<qi*<1. Moreover, qi (t, qi

0) � qi*, globally, for anyqi

0 # (0, 1).

Proof. Obviously, (0, 1) are stationary states of (3), as well as all qi

which solve d i (qi)&d j (1&qi)=0. The equation d i (qi)&d j (1&qi)=0 hasa unique solution, qi*, since, by cultural substitution, (i) d i is decreasingand d j increasing in qi, and, moreover, (ii) d i (0)>d j (1), d j (0)>d i (1).Also, (ii) implies that 0<qi*<1.

To show that qi (t, qi0) � qi*, for any qi

0 # (0, 1), note that �q* i ��qi |qi=0=d i (0)&d j (1) is >0, and �q* i��qi |qi=1=d j (0)&d i (1) is >0, if d i satisfiescultural substitution. Since 0<qi *<1 is unique, continuity of the map (3),from qi into q* i, implies that the basin of attraction of qi * is (0, 1). K

If vertical and oblique transmission are cultural substitutes, the directfamily socialization efforts of type i agents are decreasing in the fraction ofthese individuals in society, and moreover the direct socialization effort ofsmall enough a minority is larger than that of the corresponding majority,e.g., d i (0)&d i (1)>0. As a consequence the socialization pattern moves the


system away from full homogeneity: qi=0 and qi=1 are locally unstablestationary states of (3), and the basin of attraction of the unique steadystate associated to heterogeneous population, qi*, is the whole (0, 1).10

Explicit models of cultural transmission are needed to impose restrictionson the form of the function d i (qi). We will next show that several ``natural''models of endogenous cultural transmission satisfy cultural substitution,and hence are characterized by dynamics of the distribution of culturaltraits in the population which converge to an heterogeneous population.

2.2. Endogenous Cultural Transmission Mechanisms

We now study cultural transmission mechanisms in which parents takeactions to socialize their children. Such mechanisms determine endogenouslythe direct socialization maps, d i (qi), i # [a, b].

Every agents in the adult part of his lifetime makes economic and socialdecisions. We capture these decision, abstractly, with the choice of x # X tomaximize preferences ui : X � R, for i # [a, b]. Different cultural traits arethen associated to different sets of preferences over x.

Parents are altruistic towards their children and hence might want tosocialize them to a specific cultural model if they think this will increasetheir children's welfare. Let V ij denote the utility to a type i parent of a typej child, i, j # [a, b]. The expected lifetime gains (abstracting from socializa-tion costs) of a family of type i at time t are then:11

ui (x)+(Pii V ii+Pij V ij ).

Regarding parents' introspective view of their children's preferences (thedetermination of V ii, V ij), we assume that parents are only able to evaluatetheir children's actions with their own preferences. A type i parent, forinstance, is altruistic in the sense that he or she receives utility from his orher child's future socio-economic action, but the utility he or she receivesstems from the evaluation of his or her child's action with type i preferences,even if the child turns out to have type j preferences. Formally stated, theassumption has the following form.


10 Cultural substitution of direct vertical and oblique transmission is clearly much strongerthan necessary for the stability results of Proposition 1. (For instance, d i (1)=d j (1)=0 canbe substituted with d i (0)>d j (1); and the monotonicity requirement can be substantiallyrelaxed.) A weaker definition would not expand in a substantial sense though the class ofendogenous transmission mechanisms which turn out to possess the substitution property.

11 Separability of preferences between socio-economic activities and socialization substan-tially and crucially simplifies the analysis. Another important simplification we introduce isthe fact that parents do not care about the whole dynasty and do not take into accountsocialization costs which their children bear.

Assumption 1 (Imperfect empathy). For all i, j # [a, b], V ij=ui (x j ),where x j=arg maxx # X u j (x), and xa{xb.

We interpret such assumptions as a form of myopic or paternalisticaltruism (hence the name, `ìmperfect empathy''). Parents are aware of thedifferent preference traits children can adopt, and are able to anticipate thesocio-economic choice a child with preference trait i # [a, b] will (optimally)make. Parents are not able, though, to altruistically evaluate their children'sactions with the children's utility function (to ``perfectly empathize'' withthe children), but they are biased by their own (the parents') preferenceevaluations.

As a consequence of imperfect empathy, parents, while altruistic, preferchildren with their own cultural trait and hence attempt at socializing themto this trait (children with a different cultural trait will not choose the samesocio-economic action their parents would choose in their position):12

for all i, j with i{ j, V ii>V ij. (4)

Some justifications of imperfect empathy, from an evolutionary perspective,are identified by Bisin and Verdier [9].13

To analyze the socialization decisions of parents, we then need notintroduce the notation for parent's i socialization to cultural trait j =3 i; andwe denote with d i the probability of direct socialization of parents withtrait i to the i trait.

Each parent of type i can affect the probability of direct socialization ofhis child, by controlling an n-dimensional vector {i of `ìnputs''. Examplesof the elements of the inputs vector contain the time spent with the child,the cultural homogeneity of the neighborhood in which the family locates,and of the school to which the child is sent. A map D: Rn

+_[0, 1] �[0, 1] represents the ``production function'' of direct socialization:

d i=D({i, qi ). (5)

The production function D is allowed to depend on qi to capture thepossible dependence direct transmission on the distribution of the traits inthe population.


12 If the set X depends on the agent's trait, or else if agents interact strategically and hence,for instance, agents preferences depend on qi, it is not in general true that imperfect empathyimplies (4). In this paper we do restrict ourselves, however, to environments in which V ii>V ij.

13 Such an imperfect notion of altruism is also at the core of Adam Smith's Theory of MoralSentiments, where it is introduced as follows: `Às we have no immediate experience of whatother men feel, we can form no idea of the manner in which they are affected, but by conceiv-ing what we ourselves should feel in the like situation. Though our brother is upon therack, .... our senses will never inform us of what he suffers. .... By the imagination we placeourselves in his situation ....'', Part I, Section I, Chap. I.

Socialization is costly. Let C({i) denote socialization costs.

Assumption 2 (Socialization). For any i # [a, b]:

(i) the utility function ui (x) is C2, monotonic increasing and strictlyconcave, and the choice set X is convex and compact;

(ii) the map D is C2, strictly increasing and strictly quasi-concavein {i ; Moreover, D(0, qi)=0, \qi # [0, 1];

(iii) the map C is C2, strictly increasing and strictly quasi-convex;moreover, C(0)=0, and �C(0)��{i=0.

Parents of type i choose {i # Rn+ , and xi # X to maximize:

ui (x i)&C({i)+(PiiV ii+PijV ij ) s.t. 1, 2, and 5. (6)

Under Assumptions 1 and 2, the argmax of the socialization problem of atype i parent, problem (6), is represented by a continuous map which wedenote d i=d(qi, 2V i), where 2V i=V ii&V ij is the subjective utility gain ofhaving a child with trait i; it reflects the degree of ``cultural intolerance'' oftype i 's parents with respect to cultural deviations from their own trait.Given imperfect empathy on the part of parents, 2V i>0, by Eq. (4).

The dynamics of the fraction of the population with cultural trait i isthen determined by Eq. (3) evaluated at d i (qi )=d(qi, 2V i).14

The next simple Proposition develops sufficient conditions for the map dto possess the cultural substitution property.

Proposition 2. Under Assumptions 1 and 2, d(qi, 2V i ) satisfies thecultural substitution property if C and D are homothetic in {i, and

�D({i, qi)�qi �0. (7)

In particular, the cultural substitution property holds if direct socializa-tion is independent of qi.

Proof. We construct first the indirect cost function of direct socializa-tion d i, i # [a, b], which we denote H(d i, q i ):

H(d i, qi ) := min{ i # R

n+

C({i), s.t. d i=D({i, qi ). (8)

Assumptions 2(ii) and 2(iii) imply that the minimization problem in (8) isconvex, and hence: H(d i, qi ) is a continuous function in d i and qi, the


14 Note that 2V i must be chosen so that 0<d(qi, 2V i )<1, for any qi # [0, 1], sinced(qi, 2V i ) is a probability.

argmin {i is a continuous mapping from [0, 1]2 into RN, {(d i, qi ). As aconsequence then, H(d i, qi ) is convex in d i, and satisfies: H(0, qi)=0,\qi # [0, 1], and �H��d i |di=0=0.

Let *i denote the Lagrange multiplier of the constraint in Problem (8).The Implicit Function Theorem on the first order conditions of Problem(8) implies that {i (d i ) is differentiable increasing in d i, and sign (�*i��qi)=&sign (�D��qi ). But, by the Envelope Theorem, (�D��qi )=*i, and hence

sign \ �2H�qi �d i+=&sign \�D

�q i+ .

Problem (6) can now be written as a choice problem in d i :

max(x i, di) # X_[0, 1]

ui (xi)&H(d i, qi)+V ij+[(d i+(1&d i) qi] 2V i. (9)

The analysis above of the properties of H(d i, qi) guarantees that Problem(9) is convex. The Implicit Function Theorem on the first order conditionsof (9) implies then that (�d i��qi )<0 if �2H��q i �d i�0, which, we haveshown above, is satisfied if �D��qi�0. Moreover, clearly d(1)=0, since theterm [(d i+(1&d i) qi] 2V i] in the objective of Problem (9) is independentof d i at qi=1. K

For the class of endogenous socialization mechanisms introduced in thissection, then, (strong but) simple conditions on the technology of directsocialization guarantee cultural substitution, and hence the heterogeneity ofthe long run distribution of the cultural traits in the population. In par-ticular, besides convexity and homotheticity of C and D, it suffices that�D({i, qi)��qi�0. If, on the other hand, in fact D({i, qi) is increasing in q i,given {i, a form of (cultural) complementarity between direct and obliquetransmission is introduced: direct socialization is in this case more efficient,other things equal, when the trait to be transmitted is held by a majorityof the population, and hence when oblique transmission is more efficient.

While �D({i, qi)��qi�0 is clearly not a necessary condition for culturalsubstitution, and, a fortiori, it is not necessary for the stability of hetero-geneous limit distribution of the population with respect to the preferencetraits, it is easy to show that strong enough forms of cultural complementaritycan drive the dynamics of the distribution of the traits in the populationtowards homogeneity (see the example in Section 2.2.3).

The relevance of cultural substitution and complementarity in thesocialization technology is further illustrated by the examples of socializa-tion mechanisms that we proceed to analyze.

2.2.1. ``It's the family,...!'' Consider the simplest (and hence benchmark)cultural transmission technology, in which family models are the first


crucial cultural models the child is exposed to. Direct socialization is drivenonly by an effort variable, {i, and there are no interactions between societyat large and direct vertical socialization: d i=D({i)={i. The map D in thiscase then trivially satisfies Assumption 2(ii) and Condition (7). If preferencesand socialization costs satisfy Assumption 2(i) and 2(iii), then, d(qi, 2V i) isdecreasing in qi, and d i (1)=0, and, by Proposition 2, oblique and directvertical cultural transmission are cultural substitutes.15 By Proposition 1,then, the dynamics of the distribution of cultural traits converges to anheterogeneous limit distribution.

Also note that d(qi, 2V i) is increasing in 2V i=V ii&V ij, the ``degree ofintolerance'' of family i for children with cultural trait different than theparents' own. Naturally, the more `intolerant' a parent is, the larger are hisor her incentives to socialize his or her child to his or her own trait.

2.2.2. ``Do not talk to strangers.'' Consider the following socializationmechanism. Children are exposed simultaneously to their parent's trait, sayi, and to the trait of an individual picked at random from a restrictedpopulation, composed of a fraction { i

2 of agents with trait i, which can beinterpreted, for example, as the population of neighbors, peers and teachers.The parents direct socialization effort is denoted { i

1 # [0, 1], and controls thechildren's internalization of the parent's trait. If the two traits match (i.e., ifthe child internalizes his parent trait, i, and the trait of the individual in therestricted population is also i), then the child is socialized to trait i.Otherwise, with probability (1&{ i

1{ i2), the child picks a trait from the pop-

ulation as a whole. The probability that a child of a type i father is directlysocialized (by exposure to the parent and to the restricted pool) is then:

d i=D({i)={ i1 { i

2 .

Both the direct socialization effort { i1 # [0, 1], and the segregation effort,

{ i2 # [0, 1], are chosen by parents, and

{i=_{ i1

{ i2& .16

The map D, in this case also, satisfies Assumption 2(ii) and condition (7).If preferences and socialization costs satisfy Assumption 2(i) and 2(iii),


15 Another interesting specification of segregation costs consists in having them depend onthe distribution of traits in the population: C({i, qi ), with �C({i, qi )��qi>0, to capture theidea that effort and segregation are more costly for minority groups than for majority groups.It can then be shown that the results described in this section still hold provided that�2C({i, qi)��{ i

s �qi, for s # [1, 2], is not too large.16 Another interesting socialization mechanism has d i=D({i)={ i

1+(1&{ i1) { i

2 , whichcorresponds to the case in which children pick the trait of their parents from a restricted pool,with probability { i

2 , only if a first direct socialization effort, { i1 , has not been successful. While

such a direct socialization mapping D does not satisfy Assumption 2(ii), it still deliverscultural substitution in d i (qi ).

then, by Proposition 2, direct vertical and oblique transmission are sub-stitutes for such transmission mechanisms, and, by Proposition 1, thedynamics of the distribution of cultural traits converges to anheterogeneous limit distribution.

The result is driven by the fact that socialization effort { i1 and the degree

of segregation { i2 are good substitutes in the technology of direct vertical

socialization. Because of the substitutability of effort and segregationchoices, allowing for endogenous segregation choice in fact reinforces thesubstitutability of vertical and oblique transmission in the benchmarkmodel in Section 2.2.1.

2.2.3. ``It takes a village to raise a child.'' As an illustration of culturaltransmission mechanisms in which complementarities arise in vertical andoblique transmission, consider the following example. Any child is firstexposed simultaneously to the parent's trait and to the trait of a role modelfrom the population with which he or she is matched randomly. If theparent and the role model are culturally homogeneous, the child is directlysocialized to their common trait, otherwise the child is matched a secondtime randomly with a role model from the population, and adopts his orher trait. Parents of type i choose an effort variable {i. The probability thata child is directly socialized is then d i=D({i )={iqi, which does not satisfycondition (7). Vertical and oblique transmission are not cultural substitutesin this example. As a consequence, we can easily identify classes of environ-ments for which the dynamics of the distribution of traits in the populationis such that an homogeneous population in general arises in the limit.

Suppose for instance that the cost function is quadratic, C({i)= 12 ({ i)2.

In this case then, d(qi, 2V i)=(qi )2 (1&q i) 2V i ; and a simple analysis ofthe dynamics implies that for 0<qi V<1

qi (t, qi0) � 0, for any q i

0 # [0, qi *);(10)

qi (t, qi0) � 1, for any q i

0 # (qi*, 1];

see Fig. 2.17

3. WELFARE IMPLICATIONS

Without starting from some a priori ethical point of view, it is in generaldifficult to provide normative statements on economic situations with


17 Note that d i (qi ) is not always decreasing in the fraction of individuals of type i (it isincreasing for qi<2�3). This reflects the conflict between complementarities and substitutabilitiesof vertical and oblique transmission: for small qi oblique transmission is complementary. Verticaltransmission and socialization effort by parents of type i is increasing with the fraction ofindividuals of the same type.

FIG. 2. The dynamics under cultural complementarity.

evolving preferences. In the present context of endogenous socializationthough, we can characterize necessary and sufficient conditions for theefficiency of socialization patterns, d i (qi ), for i # [a, b], and the associatedpaths of the distribution of traits in the population.

Definition (Efficiency). A pattern of direct socialization, (d i (qi),d j (1&qi )), and the induced path for the distribution of cultural traits,qi (t, qi

0), are efficient if there does not exist a direct socialization pattern(d $i (qi), d $ j (1&qi )), and an induced path q$ i (t, qi

0), which are preferred to(d i (qi), d j (1&qi), qi (t, qi

0)), by both types of agents, [a, b], at each time t.

We are now ready to evaluate the efficiency of direct socialization patternswhich derive from the agents' socialization problem, Problem (6), i.e., theefficiency of (d(qi, 2V i ), d(1&qi, 2Vj )).

Proposition 3. Suppose the direct socialization patterns, (d(qi, 2Vi),d(1&qi, 2Vi)), and the induced paths of the distribution of cultural traits,qi (t, qi

0), are such that, given qi0 and for some t,

(d(qi (t, qi0), 2Vi), d(1&qi (t, qi

0), 2V j))>0. (11)

Then (d(qi, 2Vi), d(1&qi, 2V j)) and qi (t, qi0) are inefficient.


Proof. Consider the following direct socialization pattern, alternative to(d(qi, 2V i), d(1&q i, 2V j), from time 0 on: d $(qi, 2V i)=max[0, d(qi, 2V i)&d(1&qi, 2V j)] and d $(1&qi, 2V j)=max[0, d(1&qi, 2V j)&d(qi, 2V i)].The same path of the distribution of traits for t�0 is induced by the alter-native socialization pattern, qi (t, q i

0)=qi$(t, q i0). But, under the alternative

socialization patterns, costs are strictly reduced for all times t such thatd(qi (t, q i

0), 2V i), d(1&q i (t, q i0), 2V j)>0, and for both groups, thereby

proving the statement. K

Direct socialization patterns are inefficient as long as at some time t themembers of both cultural groups actively socialize their children (11).Conversely, a necessary and sufficient condition for the efficiency of directsocialization patterns is that, for all t, the members of at most one of thecultural groups actively socialize their children. Under Assumptions 1, 2and cultural substitution, in particular, (11) is satisfied for all initial condi-tions qi

0 which do not coincide with a stationary state of the system, andhence direct socialization patterns are inefficient.

It is easy to show that the direct socialization pattern (d $(qi, 2V i ),d $(1&qi, 2V j)), which we constructed in the proof of Proposition 3, notonly dominates (d(qi, 2V i), d(1&qi, 2V j)), but is also efficient. Forenvironments in which direct and oblique transmission are cultural sub-stitutes, in particular, the optimal socialization policy at any time t�0 con-sists in not allowing direct vertical socialization for the majority culturalgroup, more precisely the group i such that qi (t, q i

0)�qi*, and in reducingaccordingly the direct socialization effort of minorities, so as to reproducethe same path of distribution of traits which would have been followedfrom t=0 were the socialization policy not introduced, qi (t, q i

t), t�0.

4. CULTURAL TRANSMISSION AND EVOLUTIONARY SELECTION

The interaction between genetic and cultural transmission is at the coreof a endless cross-disciplinary debate, often referred to as the ``nature-nurture debate'' (see Rogers [31] for an illuminating introduction), whichwe will not directly address. Rather, we are interested in two quite specificquestions. First, is the cultural transmission model we introduced in theprevious section distinguishable in terms of implications from a simpleevolutionary selection mechanism? And secondly, are our results robust tothe introduction of evolutionary as well as cultural transmission mechanisms?A qualified positive answer to both questions is given for cultural transmissionmechanisms for which direct and obllque transmission are cultural substitutes.

To simplify notation, we restrict the analysis of this section to the bench-mark transmission mechanism of Section 2.2.1, in which the probability of


direct socialization coincides with an effort variable chosen by parents:d i=D({i, qi)={i ; and vertical and oblique transmission are substitutes.

4.1. Evolutionary Selection and Endogenous Fitness

Suppose the cultural transmission mechanism is as introduced in theprevious section, but the parent does not have a direct socialization choice,and, to guarantee symmetry, d i=d j=d # (0, 1). In such simple environ-ments, material payoffs, and, hence, fitness are exogenously associated witheach preference trait. Suppose that the number of children of each parent(i.e., his or her fitness) depends on the cultural group of the parent, and isdenoted ni, i # [a, b]. The dynamics of the distribution of cultural traits inthe population is determined then by

q* i=qi (1&q i) d(&i&& j ) (12)

for &i=ni�(ni+n j). It is easy to see that

0 for q i0 # [0, 1) if n j>ni

qi (t, q i0) �{1 for q i

0 # (0, 1] if ni>n j

q i0 for all q i

0 # [0, 1] if n j=ni

The dynamics of the distribution of traits in the population for the evolutionaryselection mechanisms considered tend to an homogeneous distribution,except in the degenerate case in which the cultural groups are not dis-tinguishable in terms of fitness. This conclusion sharply contrasts with theone we reached for cultural transmission models with cultural substitution(Proposition 1). In such analysis in fact, evolutionary selection mechanismslack the endogenous differential socialization success of cultural minorities,which drives the cultural heterogeneity of the stationary state distributioninduced by cultural transmission mechanisms.

It is of interest then, to compare the implications of cultural transmissionmechanisms with those of a evolutionary selection mechanisms in whichfitness is endogenously determined by parents.

Suppose each parent must choose the number of children to raise, andfaces costs c(n) to raise n children.

Assumption 3. For any i # [a, b]:

c: R+ � R+ is C2, strictly increasing andstrictly convex, c(0)=0 and c$(0)=0.


Suppose also that a parent of type i receives an utility gain V ij for eachchild of type j, i, j # [a, b].

Parents of type i then choose ni�0 and xi # X to maximize

ui (x i)&c(ni)+ni (PiiV ii+PijV ij), (13)

where Pii and Pij are defined by (2), and are evaluated at d i (qi)=d j (1&qi)=d. Let n(qi, 2V i), n(1&qi, 2V i), denote the solutions of themaximization problem (13) for parents of type i and j.

Proposition 4. Under Assumption 1, 2(i), 2(iii) and 3,

If V ii>dV jj+(1&d ) V ji, for i, j # [a, b], i =3 j : (0, 1, qi*) are thestationary states of (12) evaluated at ni=n(qi, 2V i), n j=n(1&qi, 2V i), and0<qi*<1; moreover, qi (t, q i

0) � 0, for any q i0 # [0, qi *), and qi (t, q i

0) � 1,for any q i

0 # (qi*, 1].

If instead V ii�dV jj+(1&d ) V ji for a group i # [a, b], q i (t, q i0) � 0,

globally, for any q i0 # [0, 1).18

Proof. Assumptions 1, 2(i), 2(iii), and 3 imply that problem (13) isconvex. The first order condition of the maximization problem is

(d+(1&d) qi) V ii+((1&d)(1&qi)) V ij=c$(n i)

and hence n(qi, 2V i) is differentiably increasing in qi. Any interior stationarystate, must solve n(qi, 2V i)&n(1&qi, 2V j)=0, and hence (d+(1&d ) qi) V ii

+((1&d )(1&qi)) V ij=(d+(1&d)(1&qi)) V jj+((1&d ) qi) V ji. A uniquesuch stationary state therefore exists if V ii>dV jj+(1&d) V ji, for i, j # [a, b],i =3 j. The analysis of the stability properties of the dynamics of Eq. (12),evaluated at ni=n(qi, 2V i), ni=n(1&qi, 2V j), now simply follows.

If instead, for some i # [a, b], V ii�dV jj+(1&d ) V ji, an interior steadystate does not exist, and again the stability analysis simply follows. K

Even if we allow the agents some choice of reproductive pattern (endogenousfitness), families in a cultural majority will choose relatively high fertility rates,since in this case their children will inherit their trait with high probability(children are of high expected ``quality''). The choice of reproductionpatterns, as a consequence, different from the socialization choice in anenvironment, characterized by cultural substitution (Proposition 1),induces a culturally homogeneous configuration of the distribution of traits


18 The case in which V ii�dV jj+(1&d ) V ji and V jj>dV ii+(1&d ) V ij contradicts imper-fect empath, Assumption 1.

in the population (which configuration depends on the parameters of thetransmission process, including the initial distribution of traits).

4.2. Co-Evolution

We now study the interaction between evolutionary selection and culturaltransmission. First of all, it is easy to see that the class of transmissionmechanisms which interact cultural transmission and exogenous fitnessmaintain the qualitative properties of the underlying cultural transmissionmechanism. In particular, with cultural substitution, the dynamics of thedistribution of traits converge to an heterogeneous distribution, for anyinitial condition q i

0 # (0, 1). When an exogenous fitness is associated toeach trait, the stable limit stationary distribution depends on the fitnessratio, Ri :=ni�n j, in the sense that q i* increases with Ri.

A more interesting class of transmission mechanisms has cultural trans-mission interacting with endogenous fitness. Parents of type i choose xi # X,d i # [0, 1], and ni�0 to maximize:

ui (x i)&c(ni)&niH(d i)+n i (PiiV ii+PijV ij) (14)

where Pii and Pij are as in (2).19 Let d(qi, 2V i), d(1&qi, 2V j), n(qi, 2V i),n(1&qi, 2V j), denote the solutions to the maximization problem (14) forparents of type i and j. The dynamics of the distribution of traits in thepopulation is then determined by

q* i=qi (1&q i)(d(qi, 2V i) &(qi, 2V i)&d(1&q i, 2V j) &(1&qi, 2V j)),

(15)

where &(qi, 2V i)=n(qi, 2V i)�(n(q i, 2V i)+n(1&qi, 2V j)).We will next show that, for such an environment, there exists an hetero-

geneous limit distribution of the population with respect to the preferencetrait which is locally stable. To this end we require some regularity condi-tions: (i) that parents derive enough utility from parenthood, in particular,that they would be willing to raise a (small positive fraction of a) child withpreferences different from theirs with probability one; and (ii) that thesocialization gains are large enough that parents will always want to haveat least a (small positive fraction of a) child even if constrained to directlysocialize the offspring with probability 1. Formally:


19 Following the notation introduced in the proof of Proposition 2, the choice problem ofeach parent of type i is considered as the choice of d i, with socialization costs H(d i, qi). In thesimple case considered here in which d i={i, this reduces to H(d i)=C(d i). As a consequencethe function H(.) has the same properties as C(.), from Assumption 2. Also, abusing notation,in the following we impose additional assumptions directly on H(d i) rather than on C({i).

Assumption 4. For any i # [a, b]:

(i) V ij>0,

(ii) V ii>H(1).

We can then prove:

Proposition 5. Under Assumptions 1, 2, 3, and 4, (0, 1) are stationarystates of (15). There also exists at least one stationary state qi *, such that0<qi*<1, and qi (t, q i

0) � qi*, locally, for q i0 in an open neighborhood

of qi *.

Proof. The first order conditions of problem (14) are:

V ij+[d i+(1&d i) qi] 2V i=c$(n i)+H(d i) (16)

ni (1&qi) 2V i=niH$(d i). (17)

It is easy to see that the second order conditions for a maximtim also aresatisfied. Given Assumption 2, the function 3(d i)=V ij+d i2V i&H(d i) isstrictly concave and 3 $(0)=2V i (>0 by Assumption 1). Moreover 3(0)=V ij>0 (by Assumption 4(i)), and 3(1)=V ii&H(1)>0 (by Assumption4(ii)). Thus one gets:

V ij+[d i+(1&(d i) qi] 2V i&H(d i)�V ij+d i 2V i&H(d i)=3(d i)>0

for any d i # [0, 1] and qi # [0, 1]. Hence, as c$(0)=0, it follows from (16)that n(qi, 2V i)>0, for any qi # [0, 1]. Then (17) can be solved for d(qi, 2V i),implying that d i is differentiably decreasing in qi, and d i (1)=0. As a conse-quence, �q* i��qi |qi = 0 = n(0, 2V i) d(0, 2V i) & n(1, 2V j) d(1, 2V j)>0, and�q* i��qi |qi=1=n j (0) d j (0)&ni(1) d i (1)>0, and (0, 1) are locally unstablestationary states. Continuity of the map (15), implies that at least one interiorstationary state of the system exists, solving n(qi, 2V) d(qi, 2V i) d(qi, 2V i)&d(1&qi, 2V j) n(1&q i, 2V j)=0, which is locally stable. K

While it is in general not possible to guarantee uniqueness of the interiorstationary state, qi*, for transition mechanisms which interact culturaltransmission and evolutionary selection with endogenous fitness determina-tion, we are able to provide a robust example here. (Note that, uniquenessof qi * implies its global stability: qi (t, q i

0) � qi*, for q i0 # (0, 1).)

Assume that the costs of socialization, H(d i), and the cost to raisechildren, c(ni), are quadratic and given by: H(d i)= 1

2 (d i)2, c(ni)= 12 (ni )2.


Moreover, assume V ii> 23 .20 In this case the interior steady state, qi*, is

unique.

Proof. The first order conditions for the maximization problem (14), inthe case of the example, can be written:

ni =V ii&(1&d i)(1&qi) 2V i& 12 (d i)2

d i =(1&qi) 2V i

or

ni=V ii&d i+ 12 (d i)2

d i =(1&qi) 2V i.

Let X i :=(1&qi) 2V i. The first order conditions can then be written: d i=d i (X i)=X i, and ni=ni (X i)=V ii&X i+ 1

2 (X i)2 ; and

�d i (X i) ni (X i)�X i =V ii&2X i+

32

(X i)2.

Since X i is decreasing in qi, at most one interior stationary state exists if�d i (X i) n i (X i)��X i>0. But 2X i& 3

2 (X i )2 has a maximum in X i= 23 , with

value 23 . V ii> 2

3 then is sufficient to guarantee that �d i (X i) ni (X i)��X i>0,and hence that at most one interior stationary state exists. Finally, Proposition5 implies that at least one interior stationary state exists. K

5. CONCLUSIONS

This paper presented a simple model of preference evolution based onendogenous cultural transmission. We identified conditions under whichthe long run distribution of preference traits in the population is hetero-genous. We also showed that because of natural externalities associatedwith social learning and influences between cultural groups, the path ofevolution of preferences is inefficient in the sense that, for a given dynamicprofile of preferences, too much resources are devoted to family socializa-tion. Finally, we have shown the robustness of our results when one also


20 The parameters of the parents' socialization choice are normalized so that V ii measuresthe number of children of a family with trait i, in equilibrium, when d i=0, i.e., the maximalnumber of children a family of type i would ever want to have (see the first order conditionsof the fertility choice, reported in the following Proof ). As a consequence, V ii> 2

3 is effectivelyno restriction.

incorporates in the analysis evolutionary selection mechanisms withexogenous or endogenous fitness.

Clearly, however, many aspects related to cultural transmission havebeen left out of the analysis. In particular, our analysis did not considersituations in which agents interact in socio-economic environments, nortraits which affect, in a relevant manner, the economic environment theagents face. Such extensions would be particularly fruitful to understandthe effects of market or public institutions on cultural evolution.21 Also,many important aspects of the interaction of cultural transmission andevolutionary selection mechanisms have been left out by our analysis.Studying such interactions could be useful to provide micro foundations forsocial selection mechanisms.

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